1 Mesopotamia Here We Come Lecture Two

2 Outline  Mesopotamia civilization  Cuneiform  The sexagesimal positional system  Arithmetic in Babylonian notation  Mesopotamia algebra

3 Mesopotamia (the land between the rivers) One of the earliest civilization appeared around the rivers Euphrates and Tigris, present-day southern Iraq.

4 Brief History of the “Fertile Crescent” Ishtar Gate of Babylon Persian King Darius Assyrian art 3000 – 2000 BC, Sumerians Around 1800 BC, Hammurabi 2300 – 2100 BC, Akkadian 1600 – 600 BC, Assyrians 600 – 500 BC, Babylonian 600 – 300 BC, Persian Empire 300 BC – 600 AD, Greco-Roman 600 AD -, Islamic

5 Tower of Babel Artistic rendering of “Tower of Babel” Reconstructed Ziggurat made of bricks.

6 Written System in Mesopotamia

7 Cuneiform Cuneiform tablets are made of soft clay by impression with a stylus, and dried for record- keeping.

8 The Basic Symbols 1 (wedge) 10 (chevron)

9 Base 60 (sexagesimal) = =2* =11*60+12

10 Babylonian Sexagesimal Position System 1* * * =

11 General Base b Number  A sequence represents value  Examples: b=10: = b=2: = 1*2 2 +0*2 1 +1=5 b=60: [1, 28, 52, 20] 60 = 1* * *60+20=319940

12 Babylonian Fraction (Sexagesimal Number) Fractional part

13 Conversion from Sexagesimal to Decimal  We’ll use the notation, e.g., [1, 0 ; 30, 5] to mean the value 1*60 + 0*1 + 30* *60 -2 = 60+1/2+1/720= …  In general we use the formula below to get the decimal equivalent:

14 Conversion from Decimal to Sexagesmal  Let y = a n 60 n + a n-1 60 n-1 + …, try largest n such that y/60 n is a number between 1 and 59, then y/60 n = a n + a n-1 /60 + … = a n + r  The integer part is a n and the fractional part is the rest, r.  Multiple r by 60, then the integer part will be a n-1 and fractional part is the rest. Repeat to get all digits.

15 Conversion Example  Take y = = 100+1/4  n=2, y/3600 is too small, so n=1; y/60 = 1 + (40+1/4)/60 -> a 1 = 1  r 1 =(40+1/4)/60, 60*r 1 =40+1/4 -> a 0 =40, r 0 =1/4  60*r 0 = 15, -> a -1 =15  So in base 60 is [1, 40 ; 15]

in base 60 A Better Work Sheet  /60 = … -> a 1 =1  60 x … = >a 0 =40  60 x 0.25 = … ->a -1 =15  60 x = 0 -> a -2 = 0 1* /60 =

17 Adding in Babylonian Notation + Every 60 causes a carry! = = =

18 Multiplication in Decimal 1x1=1 1x2=22x2=4 1x3=32x3=6 3x3=9 1x4=42x4=8 3x4=12 4x4=16 1x5=52x5=10 3x5=15 4x5=20 5x5=25 1x6=62x6=12 3x6=18 4x6=24 5x6=30 6x6=36 1x7=72x7=14 3x7=21 4x7=28 5x7=35 6x7=42 7x7=49 1x8=82x8=16 3x8=24 4x8=32 5x8=40 6x8=48 7x8=56 8x8=64 1x9=92x9=18 3x9=27 4x9=36 5x9=45 6x9=54 7x9=63 8x9=72 9x9=81

19 Multiplication in Sexagesimal  Instead of a triangle table for multiplication of numbers from 1 to 59, a list of 1, 2, …, 18, 19, 20, 30, 40, 50 was used.  For numbers such as b x 35, we can decompose as b x (30 + 5).

20 Example of a Base 60 Multiplication x + 51 x 25 = (1275) 10 = 21x = (21, 15) 60

21 Division  Division is computed by multiplication of its inverse, thus a / b = a x b -1  Tables of inverses were prepared.

22 Table of Reciprocals ,45451, ,20481, , ,30541,6, , ,30272,13,201,456,15 96,403021, ,52,301, ,401, ,301,2144,26,40 aa -1 aa -1 aa -1

23 An Example for Division  Consider [1, 40] ÷ [0 ; 12]  We do this by multiplying the inverse of [0 ; 12 ]; reading from the table, it is 5.  [1, 40] × [5 ; 0] = [5, 200] = [8, 20]

24 Sides of Right Triangles 90 ° a b c In a clay tablet known as Plimpton 322 dated about 1800 – 1600 BC, a list of numbers showing something like that a 2 + b 2 = c 2. This is thousand of years before Pythagoras presumably proved his theorem, now bearing his name.

25 Plimpton 322 line number cb(c/a) 2 a 2 + b 2 = c 2, for integers a, b, and c <- line 11 Line number 11 read (from left to right), [1?; 33, 45], [45], and [1,15]. In decimal notation, we have b = 45, c=75, thus, a = 60, and (c/a) 2 =1 + 33/ /3600 = (5/4) 2

26 Square Root YBC 7289 The side of the square is labeled 30, the top row on the diagonal is 1, 24, 51, 10; the bottom row is 42, 25, 35.

27 Algorithm for Compute 1.Starting with some value close to the answer, say x =1 2.x is too small, but 2/x is too large. Replace x with the average (x+2/x)/2 as the new value 3.Repeat step 2 We obtain, in decimal notation the sequence, 1, 1.5, …, , …

28 Word Problem (Algebra)  I have multiplied the length and the width, thus obtaining the area. Then I added to the area, the excess of the length over the width: 183 was the result. Moreover, I have added the length and the width: 27. Required length, width, and area? This amounts to solve the equations, in modern notation: From Tablet AO8862, see “Science Awakening I” B L van der Waerden

29 The Babylonian Procedure = 210, = 29 Take one half of 29 (gives 14 ½) 14 ½ x 14 ½ = 210 ¼ 210 ¼ = ¼ The square root of ¼ is ½. 14 ½ + ½ = 15 -> the length 14 ½ - ½ - 2 = 12 -> the width 15 x 12 = 180 -> the area.

30 Here is what happens in modern notation xy+(x-y)=183 (1), x+y=27 (2) Add (1) & (2), we get xy+x-y+x+y=x(y+2)=210. Let y’=y+2, we have xy’=210, thus x+y’=x+y+2=29 (3) So (x+y’)/2 = 14 ½, square it (x 2 +2xy’+y’ 2 )/4=(14 ½ ) 2 =210 ¼. Subtract the last equation by xy’=210, we get (x 2 -2xy’+y’ 2 )/4 =210 ¼ = ¼, take square root, so (x – y’)/2 = ½, that is x-y’=1 (4) Do (3)+(4) and (3)-(4), we have 2x= 29+1, or x = 30/2=15 And 2y’ = 29-1 = 28, y’=14, or y = y’-2=14-2 = 12

31 Legacy of Babylonian System Our measurements of time and angle are inherited from Babylonian civilization. An hour or a degree is divided into 60 minutes, a minute is divided into 60 seconds. They are base 60.

32 Summary  Babylonians developed a base 60 number system, for both integers and fractions.  We learned methods of conversion between different bases, and arithmetic in base 60.  Babylonians knew Pythagoras theorem, developed method for computing square root, and had sophisticated method for solving algebraic equations.